A single particle has no temperature. It has a certain energy or a certain speed, but it is not possible to translate this into a temperature. It is only when dealing with random velocity distributions of many particles that a well-defined temperature emerges.

How can the laws of thermodynamics derive from the laws of quantum physics? This is a subject that has been attracting more and more attention in recent years. At TU Wien (Vienna), this question has now been further explored with computer simulations, which have shown that chaos plays a crucial role: Only where chaos prevails do the well-known rules of thermodynamics follow from the quantum physics.

**Boltzmann: Everything is possible, but it may be unlikely**

Air molecules flying randomly around a room can assume an unimaginable number of different states: different locations and different speeds are allowed for each individual particle. But not all of these states are equally probable.

“Physically, it would be possible for all the energy in this space to be transferred to a single particle, which would then move at extremely high speeds while all the other particles remained stationary,” explains Professor Iva Brezinova from the Institute of theoretical physics. at TU Vienna. “But it’s so unlikely that it will hardly ever be observed.”

The probabilities of the different allowed states can be calculated according to a formula that the Austrian physicist Ludwig Boltzmann established according to the rules of classical physics. And from this probability distribution, the temperature can also be read: it is only determined for a large number of particles.

**The whole world as one quantum state**

However, this poses problems when it comes to quantum physics. When a large number of quantum particles are in play at the same time, the equations of quantum theory become so complicated that even the best supercomputers in the world have no chance of solving them.

In quantum physics, individual particles cannot be considered independently of each other, as is the case with classical billiard balls. Each billiard ball has its own individual trajectory and its own individual position at all times. Quantum particles, on the other hand, have no individuality – they can only be described together, in one large quantum wavefunction.

“In quantum physics, the entire system is described by a single large multi-particle quantum state,” explains Prof. Joachim Burgdörfer (TU Wien). “How a random distribution and therefore a temperature should derive from it has long remained a puzzle.”

**Chaos theory as a mediator**

A team from TU Wien has now been able to show that chaos plays a key role. To do this, the team performed a computer simulation of a quantum system made up of a large number of particles – many indistinguishable particles (the “heat bath”) and one particle of a different type, the “particle sample” which acts as a thermometer.

Each individual quantum wave function in the grand system has a specific energy, but no well-defined temperature, just like a single classical particle. But if you now choose the sample particle of the single quantum state and measure its velocity, you can surprisingly find a velocity distribution that corresponds to a temperature that fits the well-established laws of thermodynamics.

“Whether it is suitable or not depends on the chaos – this is what our calculations have clearly shown”, says Iva Brezinova. “We can specifically modify the interactions between particles on the computer and thus create either a completely chaotic system or one that shows no chaos – or anything in between.” And in doing so, it is found that the presence of chaos determines whether or not a quantum state of the sample particle displays a Boltzmann temperature distribution.

“Without making assumptions about random distributions or thermodynamic rules, thermodynamic behavior follows from quantum theory in its own right – if the combined system of sample particles and heat bath behaves in a quantum chaotic manner. And in how far this behavior corresponds to the well-known formulas of Boltzmann is determined by the force of chaos,” explains Joachim Burgdörfer.

This is one of the first cases in which the interaction between three important theories has been rigorously demonstrated by multi-particle computer simulations: quantum theory, thermodynamics and chaos theory.

The research is published in the journal *Entropy*.

**More information:**

Mahdi Kourehpaz et al, Canonical Density Matrices from Eigenstates of Mixed Systems, *Entropy* (2022). DOI: 10.3390/e24121740

Provided by Vienna University of Technology

**Quote**: How chaos theory mediates between quantum theory and thermodynamics (2022, December 14) retrieved December 15, 2022 from https://phys.org/news/2022-12-chaos-theory-quantum-thermodynamics.html

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